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Question 25244: Evenness and the Density Ratio Follow from the Squared Argument

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0049-25244 · q-20261015-advance-20261114-r04-standard-normal-distribution-0044

Original question and givens

Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=19/20 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).

  • G001 — Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=19/20 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [normaldist-evenness-ratio]

After

derived = [[x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate]]

Conditions: none

List the full source field order without JavaScript.LP boundary modes all have6 fields but select different roles from the original mode,not answer shape.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

Gaussian ORIGINAL x squared := mul(19/20, 19/20)

After

Gaussian ORIGINAL x squared = 361/400

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

Gaussian square times exact half := mul(361/400, 1/2)

After

Gaussian square times exact half = 361/800

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

source roles = [19/20, -361/800]

After

derived = [(2*pi)^(-1/2)*exp(-361/800), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization is a NAMED Gaussian integral premise]

Conditions: none

Exact rational arithmetic computes only the exponent−x^2/2;retain exp and its normalization factor symbolically.The whole Gaussian law and normalization are declared premises,not proved by one density value.

Step identifiers and dependencies

Rule: Gaussian_WHOLE_declared_density_symbolic_value_named_normalization · Uses: G001, S003

S005 · calculation

Before

Gaussian negative x squared := mul(-19/20, -19/20)

After

Gaussian negative x squared = 361/400

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

source roles = [19/20, -19/20, 361/400, 361/400]

After

derived = [(2*pi)^(-1/2)*exp(-361/800), (2*pi)^(-1/2)*exp(-361/800), Yes, phi(x)/phi(0)=exp(-361/800)]

Conditions: none

Equality(−x)^2=x^2 gives evenness.Phi-density phi(0)=(2pi)^−1/2 is positive,so division cancels the factor and leaves exp(−x^2/2),not a CDF or normalization evaluation.

Step identifiers and dependencies

Rule: Gaussian_evenness_and_ratio_cancel_positive_phi0 · Uses: G001, S005

Reverse checks

  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Recompute Gaussian negative x squared with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Recompute Gaussian square times exact half with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute Gaussian ORIGINAL x squared with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(19/20, -19/20, (2*pi)^(-1/2)*exp(-361/800), (2*pi)^(-1/2)*exp(-361/800), Yes, phi(x)/phi(0)=exp(-361/800), evenness_follows_from_x_squared)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 6995e3c18e3bb55ec061a7175c228b79c741beea4f5216f66fbec20ff3f8cf54
  • candidate record SHA-256: 2cd745354bd910569f4f2deb1223c16928efe512fc89503d1783fad10671658e
  • semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3
  • question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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